Two kinematic models of the radula/odontophore of the marine mollusc Aplysia californica were created to characterize the movement of structures inside the buccal mass during the feeding cycle in vivo. Both models produce a continuous range of three-dimensional shape changes in the radula/odontophore, but they are fundamentally different in construction. The radulacentric model treats the radular halves as rigid bodies that can pitch, yaw and roll relative to a fixed radular stalk, thus creating a three-dimensional shape. The odontophore-centric model creates a globally convex solid representation of the radula/odontophore directly, which then constrains the positions and shapes of internal structures. Both radula/odontophore models are placed into a pre-existing kinematic model of the I1/I3 and I2 muscles to generate three-dimensional representations of the entire buccal mass. High-temporal-resolution, mid-sagittal magnetic resonance(MR) images of swallowing adults in vivo are used to provide non-invasive, artifact-free shape and position parameter inputs for the models. These images allow structures inside the buccal mass to be visualized directly, including the radula, radular stalk and lumen of the I1/I3 cavity. Both radula-centric and odontophore-centric models were able to reproduce two-dimensional, mid-sagittal radula/odontophore and buccal mass kinematics,but the odontophore-centric model's predictions of I1/I3, I2 and I7 muscle dimensions more accurately matched data from MR-imaged adults and transilluminated juveniles.

Animal behavior is a complex interaction between neural control and peripheral biomechanics. Nervous systems have evolved simultaneously with bodies, and both develop and grow together during the lifetime of an individual animal. It is likely that animal nervous systems readily exploit all available physical properties of the periphery to simplify neural control(Chiel and Beer, 1997). Unlike human-engineered devices, which are designed to be functionally decomposable(for ease of both maintenance and integration with existing devices), animals have complex, multi-functional components. The same anatomical parts of an animal may be used in completely different ways, depending upon circumstances. Experimental observations of neural function may not be interpretable unless considered in conjunction with the properties of the body being controlled,and the converse is equally true.

Feeding behavior, which includes both appetitive and consummatory components under the control of motivational variables, requires a continuous interaction between the nervous system and the periphery during its execution and is therefore well-suited for the study of the interplay between neural control and the periphery. The feeding cycle of the marine mollusc Aplysia californica is one of several tractable model systems. The buccal mass of A. californica has a basic repertoire of movements that can be combined in several ways to create distinct behaviors at the organismal level,depending upon sensory feedback. The tongue-like radula/odontophore is capable of anterior and posterior movement (protraction and retraction) as well as opening and closure of the radular halves. These movements are caused by contractions of various muscles within the buccal mass. The relative timing between the protraction/retraction and opening/closure behaviors determines whether attempts are made to grasp prospective food (biting), which is either swallowed if edible or rejected if inedible or noxious(Church and Lloyd, 1994; Morton and Chiel, 1993a).

The buccal mass of A. californica undergoes characteristic shape changes during the feeding cycle. In two-axis studies of transilluminated juveniles in vivo, Drushel et al.(1997) found that the buccal mass could assume three distinct classes of shape (in lateral view), which correlated with three distinct phases of the feeding cycle: round (peak protraction), ovoid (neutral `rest' and transition) and Γ-shaped (peak retraction). The shapes, which are outlines of the entire buccal mass, allow some inferences about the arrangement and function of internal structures. For example, the vertical part of the peak-retraction Γ shape contains the protruding radular stalk cartilage (which must also be positioned vertically),whereas the horizontal part is the closed I1/I3 muscle and jaw cartilage complex. Viewed dorsally, the buccal mass at peak retraction is elongated antero-posteriorly and narrow medio-laterally. In contrast, the round peak-protraction shape represents the radula/odontophore protruding into the I1/I3 halves, and the radular stalk, deep inside the buccal mass, does not extend beyond the smooth outline formed by the I1/I3, I4 and I2 muscles. Viewed dorsally, the buccal mass at peak protraction is foreshortened antero-posteriorly and much wider medio-laterally. These kinematics suggest that medio-lateral contraction of the I1/I3 muscle could squeeze the radula/odontophore posteriorly, causing retraction and possibly assisting with radular closure.

Kinematic modeling may be used to test more elaborate hypotheses about muscle functions in the A. californica buccal mass, especially for internal muscles that cannot be observed in transilluminated animals or in in vitro preparations without dissection (which disrupts potentially critical interconnections between muscles). In a kinematic model, anatomical structures are represented as separate objects that are connected, move and change shape/dimensions according to anatomical rules; no forces (kinetics)are involved. Qualitatively good results could be achieved by representing the radula/odontophore as a fixed-diameter sphere that could be rotated and translated anteriorly and posteriorly through a stack of six tori(representing the I1/I3 muscle), the sphere being attached ventrally to the first torus in the stack, and the tori each remaining isovolumetric during deformation (Drushel et al.,1998). These approximations allowed the model to be completely analytical and to run rapidly on contemporary computers. The overall buccal mass shapes generated by a spherical radula/odontophore, however, proved quantitatively inaccurate (especially for peak retraction) compared with data from transilluminated juveniles (Drushel et al., 1997), so a revised, non-analytical model was developed that used in-vitro-derived volumetric databases to represent the radula/odontophore at different points in the feeding cycle(Drushel et al., 1998). No single fixed radula/odontophore shape was suitable through an entire feeding cycle (approximately 110 ° of rotation), but fixed shapes worked in that part of the feeding cycle from which they had been derived (e.g. the peakretraction shape was good in the retraction part of the cycle, but failed in the protraction part). Even though the second model was successful only in portions of the feeding cycle and took much longer to run, it allowed predictions of the lengths of the I1/I3 and I2 muscles and suggested periods of activation for the muscles that were consistent with existing neural and electromyographic recordings.

These results suggest that an accurate, three-dimensional kinematic model of the radula/odontophore, capable of smooth transformations between the round, ovoid and Γ shapes, might be developed that could then be inserted into the existing kinematic model of the buccal mass. Detailed three-dimensional kinematic analyses of the radula/odontophore in vivo and in vitro are necessary to provide the necessary input parameters and constraints for such a model (Neustadter et al., 2002a,b). Since the radula/odontophore shape is critical for the overall shape of the buccal mass model, the same qualitative and quantitative constraints used previously will serve to validate the new model — if the modeled radula/odontophore yields an inaccurate buccal mass shape (in all three dimensions), there is likely to be some flaw in the modeled radula/odontophore. In addition, the new radula/odontophore model can have its own anatomically derived internal structure, which allows predictions to be made about the shape and position of the component parts and leads to new hypotheses about their functions during the feeding cycle.

Kinematic modeling is an iterative process, and failure is as informative as success. The present paper describes a sequence of kinematic models, each of which represents some improvement upon its predecessor, each developed in response to specific predictions that were tested experimentally or against higher-resolution data. The first model of the radula/odontophore is radula-centric (where the radula and other internal components directly specify the overall shape) and the second is odontophore-centric (where the radula/odontophore is modeled as a closed surface `skin' and internal components are constrained to be inside). Both models make use of magnetic resonance (MR) imaging data from isolated buccal masses (high spatial resolution) and feeding animals (real-time, high temporal resolution; Neustadter et al., 2002a). Our ultimate goal is a complete description of the interplay between neural control and peripheral biomechanics in the feeding behavior of A. californica.

The kinematic models and analyses presented here rely heavily upon prior published work from our laboratory (Drushel et al., 1997, 1998). Since both magnetic resonance imaging (MRI) data and kinematic modeling methods are novel to the field of molluscan feeding, we have chosen to present many details to give the reader ample means to evaluate our results critically. We recognize that a long paper is the result.

As adjuncts to the text, we provide digital movies of swallowing in A. californica (in QuickTime format), three of the radula-centric model and three of the odontophore-centric model: the movie entitled `' shows the midsagittal MRI sequence 5385-S1/116-133. The movies entitled `',`' and `' show three-dimensional renderings of the radula-centric model from antero-posterior, dorso-ventral and latero-medial viewpoints, respectively, which are composites based upon MR sequences such as those shown in `'. These three movies are rendered in a perspective view. The movies entitled `', `' and `' show three-dimensional renderings of the odontophore-centric model from dorso-ventral, latero-medial and oblique viewpoints, respectively, to match the MR images in `'. These three movies are rendered in orthographic projection. Further technical details are given in the first frame of each movie. Movies of the radula-centric and odontophore-centric models are not the final kinematic models discussed in the text, but are qualitatively similar. They were created for preliminary presentations of this work that have appeared in abstract form(Chiel et al., 1999; Sutton et al., 2000).

Here, we describe the acquisition and analysis of static and real-time MR images of A. californica swallowing, the mathematical construction of the two radula/odontophore models and the in vivo measurements used to validate the output of the models.

Animals and anatomical methods

Adult Aplysia californica Cooper were obtained from Marinus (Long Beach, California, USA) and maintained in tanks of aerated artificial sea water at 16 °C. Animals used for anatomical studies were anesthetized by injection of isotonic aqueous MgCl2 (333 mmol l-1) into the hemocoel. Buccal masses were dissected out and either studied immediately or fixed overnight in 10% (v/v) formalin in Aplysia saline, pH 7.5.

Feeding cycle landmark timing nomenclature

Feeding cycle landmark timing intervals are used as defined for transilluminated juvenile A. californica in(Drushel et al., 1997): t1 (peak protraction to peak retraction), t2 (peak retraction to loss of Γ shape), t3 (loss of Γ shape to start of anterior buccal mass movement) and t4 (start of anterior buccal mass movement to peak protraction). In the present work, however, we have chosen to begin the feeding cycle with t4 rather than t1. Mid-sagittal MR images allow the precise onset of protraction to be observed because the radula/odontophore is directly visible (unlike in transilluminated juveniles, in which peak protraction and retraction were the most unambiguous landmarks), and studies have shown that all feeding responses in A. californica are initiated with protraction(Hurwitz et al., 1996). In addition, beginning at t4 allows the most variable interval, t3 (which probably corresponds to the inter-swallow interval), to be placed at the end of the feeding cycle (see Table 1 of Drushel et al., 1997). For consistency with published work, we retain the t1-t4 nomenclature without renumbering.

MRI data acquisition and analysis

A novel real-time MRI interface, imaging tank and coil and continuous feeding apparatus were developed to visualize swallowing behavior in adult A. californica (Neustadter et al., 2002a). Briefly, an Elscint 2T Prestige whole-body MRI system was used to acquire high-temporal-resolution MR images (250 ms or 310 ms per frame) with an acquisition matrix of 64×128 voxels (each voxel 1.0 mm×1.0 mm×3.0 mm). The primary aim was to study the mid-sagittal plane, since many movements and rotations of the buccal mass and radula/odontophore (e.g. protraction and retraction) are confined to this plane. To allow animals to feed continuously with a minimum of body movement,two different types of feeding stimulus were used: (i) polyethylene tubes,1.27 mm outside diameter, 0.86 mm inside diameter, after stimulation of the lips with aqueous seaweed extract; and (ii) seaweed-flavored spaghetti noodles wound on a spool in the imaging tank. Orthogonal images in the axial and coronal planes were used to verify that the animals were correctly oriented for mid-sagittal imaging (i.e. were not twisted or skewed out of the imaging plane).

One anesthetized buccal mass was imaged at high spatial resolution (each voxel 0.3 mm×0.3 mm×1.0 mm, total acquisition time 96 s per slice)for anatomical studies. These images were used to establish a consistent set of dimensions (and volumes) for structures to be modeled.

Grayscale TIFF images of MRI frames were assembled into digital movies in QuickTime format (Apple Computer). Measurements and other observations were made directly from the movie frames. The ability to switch rapidly back and forth between adjacent frames is crucial for detecting subtle movements of internal structures. Canvas 3.5.3 (Deneba Software) was used to determine the coordinates of extrema and control points in MR images used for the odontophorecentric model.

Buccal mass kinematic model

A slightly revised version of the second (non-analytical) kinematic model described in Drushel et al.(1998) was used. Briefly, the model consists of a three-dimensional representation of the radula/odontophore, which is rotated anteriorly and posteriorly through a stack of roughly concentric rings that represent the I1/I3 muscle complex. Each ring changes its dimensions isovolumetrically to conform to the shape of the underlying radula/odontophore shape, with the added constraint that ring closure occurs medio-laterally (to model the effect of the jaw cartilages). The radula/odontophore is joined to the ventral first I1/I3 muscle ring at a`hinge'; it is about this point that radula/odontophore rotation takes place. The I2 muscle is represented as a thin band wrapping around the posterior end of the radula/odontophore and connecting the dorsal and ventral portions of the first I1/I3 muscle ring.

Radula/odontophore shapes generated by the radula/odontophore models (see below) were converted to the volumetric database format required by the buccal mass model. The buccal mass model was modified to run a sequence of multiple volumetric databases at one rotation step each (the 1998 version of the model rotated a single volumetric database through multiple steps). The ability to specify individual I1/I3 ring parameters was also added (the published model made each ring a fixed-scaled reduction of the previous ring). The core functionality of the model, however, is unchanged from the 1998 version. The model computes the length of the I2 muscle and the dimensions of the individual I1/I3 rings.

The model was implemented in Microsoft QuickBASIC 4.5 for MS-DOS with SVGA graphics and extended memory libraries (Zephyr Software) When run on a Pentium III/450 MHz computer in a full-screen MS-DOS session (under Microsoft Windows 98), each rotational step of the model took approximately 3 min. The source code is available from the authors upon request.

POV-Ray 3.1 for Windows (The Persistence of Vision Development Team) was used to create ray-traced three-dimensional images of model outputs. Several utility programs were written in QuickBASIC to convert model outputs into POV-Ray-compatible format.

Radula/odontophore modeling strategies

We considered two approaches for designing a kinematic model of the radula/odontophore: inside-out versus outside-in. An inside-out model specifies the desired internal components, defines rules for their attachments, movements and dimensional changes and then stretches a `skin'over them to create a closed three-dimensional volume. An outside-in model uses rules derived from available biological data to specify a closed three-dimensional volume in its entirety; desired internal components are then constrained to fit inside this volume. In both cases, the final object is converted into the volumetric database format required by the buccal mass kinematic model, at the appropriate scale.

Radula-centric model

As a first step towards a detailed kinematic model of the radula/odontophore, we hypothesized that radular movements were the chief determinant of overall radula/odontophore shape. An inside-out model was designed around two radular halves that would be capable of opening and closure; hereafter, this is termed the radula-centric model(Fig. 1). The model consists of two radular halves, a radular stalk and two I7 muscles. Analytical functions are used to model all objects for fast computation.

The curved dorsal surface of each radular half is modeled using quadrants of ellipses and segments of parabolas, and a small portion of the flat inner cleft surface was included (Fig. 1A-C). The radular halves are attached anteriorly and can open and close by a combination of yaw (opening in the horizontal plane,θ y) and roll (rotating inwards or outwards upon their long axis, θr). The radular halves can also pitch up and down(θp) in the mid-sagittal plane; the pivot point is above the top of the radular stalk. (Empty space was left to allow for unmodeled soft tissues.) These motions were decided upon after extensive manipulation of dissected radula/odontophores from anesthetized buccal masses. The radular halves are treated as rigid bodies and do not change any of their dimensions as they move.

The radular stalk itself is a cylinder that is flat dorsally and has a rounded ventral dome (Fig. 1A). It is represented as a rigid body of fixed dimensions.

The I7 muscles are two cylinders, each parasagittal, whose long axes extend antero-dorsally from a point on the anteroventral radular stalk to a fixed point in space that represents the I7 muscles' attachment to the radular surface. This is the skirtlike portion of the radular surface that extends ventrally from the point at which the radular cleft begins. The position of the point is computed as a fixed arc length of an ellipse connecting the anterior tip of the radula and the ventral dome of the radular stalk. As the radular halves pitch, yaw or roll, the I7 endpoints change(Fig. 1C, arrowhead). The actual I7 cylinders stop slightly short of the endpoints in an attempt to avoid overlapping of the muscle objects with their attachments at shallow angles of attachment. The radius of the I7 cylinders is adjusted such that the volume of each cylinder remains constant. The antero-ventral attachment of the I7 muscles is used as the hinge point for the buccal mass model; this is where the radula/odontophore is attached to the ventral part of the first I1/I3 ring.

The dimensions of all objects were chosen to match a scale clay model of the radula/odontophore, which had been constructed for reference on the basis of several dissections of unfixed buccal masses. This model was presumed to represent a `rest' state. The overall validity of the clay model was confirmed by subsequent high-spatial-resolution MR imaging of anesthetized buccal masses in vitro. Values and timing for the radular stalk and pitch angles(Fig. 1E,F) were determined from a study of real-time MR image sequences taken from swallowing animals. Values and timing for the radular yaw and roll angles(Fig. 1G) were estimated from color VHS videotaped sequences of biting in adult animals (30 frames s-1) and from coronal and axial MR images.

The model was implemented in QuickBASIC and run in the computer environment described above. An interactive program was used to set the pitch, yaw and roll angles of the radular halves, one frame at a time. The objects could be rotated freely about any axis in real time for inspection. When each frame was complete, it was exported as a volumetric database by slicing through it in the axial plane (perpendicular to the antero-posterior axis). An elliptical skin was stretched over the objects in each slice (in two ventral quadrants),to fill in the empty spaces (presumably the I4, I5 and I6 muscles) and create a solid object, as each slice was exported(Fig. 1D). The length and radius of the I7 muscle were computed by the model. Not counting time to enter the parameters, the generation of each frame's volumetric database took approximately 5 min.

Odontophore-centric model

Even if radular movements are the primary determinant of overall radula/odontophore shape, it is possible that specific shapes may arise from interactions between multiple internal structures, including some not represented in the radula-centric model. To test this hypothesis, an outside-in model was designed to specify an entire radula/odontophore object whose volume would be kept constant during dimensional changes; hereafter,this is termed the odontophore-centric model(Fig. 2). A radular stalk and two I7 muscles are positioned inside this volume at defined connection points. For fast computation, analytical functions are used to model the radula/odontophore and the I7 muscles. The radular stalk has a realistic but fixed shape derived from high-spatial-resolution MR images of an anesthetized buccal mass.

The radula/odontophore is constructed in octants, each bounded in three planes by superellipse curves. The superellipse function, given parametrically as:
\[\ x=a\mathrm{cos}^{n}{\theta},\]
\[\ y=b\mathrm{sin}^{n}{\theta},\]
for a>0, b>0, 0≤θ≤2π and 0≤n≤∞, can generate curves of varying convexity or concavity, roundness or flatness depending upon the value of the nparameter (Fig. 2A). Parameters a and b are equivalent to the major and minor axes,respectively, of a standard ellipse. A mid-sagittal MR image is used to constrain four superellipse curves to best represent the outline of the radula/odontophore in that image (Fig. 2B). The coordinates of the anterior, posterior, dorsal and ventral extrema are selected manually; these determine the superellipse quadrant boundaries. The ventral extremum was defined to be the same as the ventral vertex of the radular stalk; this provided a place of attachment for the stalk. Four additional surface points per quadrant are chosen manually to control the curvature. The `best' value of the n parameter is then determined for each quadrant using an algorithm that minimizes the sum of distances of closest approach of the superellipse curve to the control points.

Because the radula/odontophore consists of symmetrical right and left halves, only four more out-of-plane superellipse curves are needed to create a complete three-dimensional shape, as well as the maximum medio-lateral width for the radula/odontophore (Fig. 2D,E). The default value n=1.0 (ellipse) is used for the four out-of-plane curves because there are no real-time MRI data available in the third dimension to suggest any different value. (Note that high-spatial-resolution MRI data are available for anesthetized buccal masses,but qualitative examination of their odontophores indicated that their shapes could be characterized by values of n that did not differ significantly from 1.0.) All radula/odontophores are assumed to be globally convex shapes (0≤n≤2) to simplify volume computations (see below).

We hypothesize that, because it is constructed mainly of water-containing muscle and cartilage, the radula/odontophore is a muscular hydrostat(Kier and Smith, 1985); hence,the radula/odontophore model must be isovolumetric with dimensional changes over time. For any MRI-constrained model frame, the medio-lateral width is the free parameter that is varied until the volume is equal to some predetermined constant. This standard volume is determined from the high-spatial-resolution MR images of anesthetized buccal masses and scaled between different animals using the height of the radular stalk. Total volume is computed by decomposing the model (which is created as a mesh of surface triangles) into tetrahedra,using the centroid of the model as one vertex and the vertices of each surface triangle to define each tetrahedron, and summing the volumes of each tetrahedron (calculated by analytical formula; Bronshtein and Semendyayev,1985). This method is valid only if the radula/odontophore model is globally convex; this is guaranteed by appropriate choice of the nsuperellipse parameter.

Note that radular opening and closure are not explicitly represented by this model because there are no anatomical structures present to simulate it. This does not present a difficulty to the buccal mass model (which is affected only by the maximum width), but if the actual cleft created during radular opening in vivo is large, then the odontophore-centric radula/odontophore model may not become physiologically wide enough (for example, during the protraction phase of swallowing) because the radula/odontophore is modeled as a globally convex solid and where the cleft ought to be will be `filled in' by the volume calculator.

The radular stalk model is a mesh of surface triangles created from the smoothed, stacked outlines of the radular stalk as seen in adjacent horizontal sections of high-spatial-resolution MR images of an anesthetized buccal mass. The height of the radular stalk is used as a scale bar between different animals. The radular stalk is positioned inside the radula/odontophore such that the ventral vertex of the stalk (i.e. the dome) is connected to the ventral vertex of the radula/odontophore(Fig. 2C). It does not change size or shape during the model feeding cycle.

The I7 muscles are two parasagittal cylinders (as in the radula-centric model) with conical end-caps, the tips being the attachment points. This was an attempt to model the elastic tissue at the attachment points and to avoid problems observed in the radula-centric formulation (see Results). The I7 muscles extend antero-dorsally from an anatomically defined point on the antero-ventral radular stalk to a defined endpoint on the dorsal radula/odontophore surface. This endpoint is selected from each mid-sagittal MRI frame by drawing a line from the ventral I7 attachment point along the trajectory of the muscle, through its actual insertion on the inner(posterior) surface of the I6 muscle and outwards to the radula/odontophore surface (Fig. 2B). Since this model is outside-in, the model does not represent the thickness of the I6 muscle, so it uses the corresponding radula/odontophore surface point (which is usually close to the anterior tip of the radula) as an approximation. A fixed-arc-length approximation similar to that used in the radula-centric model was tested but was too inaccurate in both length and trajectory (data not shown). As the endpoints of I7 change, the radius is changed to maintain a constant volume (cylinder + both end caps).

The dimensions of all objects were chosen with reference to the static and real-time MR images, then scaled through the radular stalk height to facilitate comparison with the radulacentric model output.

The model was implemented in QuickBASIC and run in the computer environment described above. The interactive features of the radula-centric model program were retained, but an unattended batch-processing mode was added. The length and radius of the I7 muscle and the maximum medio-lateral width of the radula/odontophore were computed by the model. The generation time for each frame's volumetric database was approximately 10 min.

Validation of model output

Shape space analysis

By approximating the mid-sagittal outline of the buccal mass as four ellipse quadrants and defining the shape parameters ellipticity (how elliptical or circular a shape is) and eccentricity (how skewed or symmetrical a shape is), it was possible to quantify the shape changes observed in the buccal mass during the feeding cycle in transilluminated juveniles(Drushel et al., 1997). Plots of eccentricity versus ellipticity form a two-dimensional shape space through which the buccal mass travels during the feeding cycle(Fig. 3). This analysis was subsequently used to evaluate the performance of two kinematic models of swallowing (Drushel et al.,1998). In the present study, shape space analysis was performed upon mid-sagittal MR images and upon model output frames, with the presumption that successful models would accurately reproduce the shape space plots of the MRI swallowing sequences upon which they were based. Manual tracings were made of the outlines of the buccal mass (MRI or modeled), converted to grayscale TIFF images of solid black shapes and median-filtered with a radius of 15 pixels using Adobe Photoshop (versions 4.0-5.5, Adobe Systems), and the ellipticity and eccentricity parameters were then determined by a computer analysis program.

Measurements from dorso-ventral views of transilluminated juveniles

Dorso-ventral views of swallowing in transilluminated juveniles, originally obtained to ensure that medio-lateral views were parallax-free(Drushel et al., 1997),contained important information about the change in medio-lateral width of the buccal mass during the feeding cycle, including changes in the width of the I1/I3 muscle complex. The length and width changes of I1/I3 were measured in three consecutive transilluminated swallowing sequences(Fig. 4). Because of uncertainty in determining the lateral borders of I1/I3, three of us (G.P.S.,E.V.M. and B.W.A.) independently performed the width measurements, and all three data sets were averaged. For purposes of comparison with buccal mass models, in which the I1/I3 muscle complex is represented as discrete rings,the I1/I3 region in vivo was divided into six segments of equal antero-posterior thickness, and the medio-lateral width of each of these`rings' was measured. This method provided more spatial detail about the in vivo shape changes and did not assume, for example, that there is a linear change in width from anterior to posterior. The medio-lateral changes in modeled I1/I3 rings were compared with these in vivo data.

Here, we present kinematic measurements from transilluminated juveniles and MR images of adult animals as well as detailed predictions of buccal mass and radula/odontophore three-dimensional shapes and muscle lengths made by the two kinematic models. Swallows of both polyethylene tubes and seaweed noodles are analyzed and modeled.

Comparison of transilluminated and MR images

The in vivo kinematics of the buccal mass during swallowing in A. californica was first described in juveniles using the non-invasive technique of transillumination(Drushel et al., 1997). The outline of the transilluminated buccal mass constrains the position of internal structures, but the exact positions and shapes of the internal structures can be deduced only at some points in the feeding cycle (e.g. the radular stalk at peak retraction). High-temporal-resolution mid-sagittal MRI of swallowing in adults is non-invasive and can show not only the buccal mass outline but also the exact positions and shapes of structures inside the buccal mass. This yields more accurate parameters for kinematic models of the radula/odontophore and enables comparison of buccal mass kinematics in adults and juveniles in vivo.

Buccal mass shapes

The outer shapes of the buccal mass are very similar between mid-sagittal MR images of adults and latero-medial transilluminated juveniles(Fig. 5). The distinctive ovoid(shape 2, transition/rest), round (shape 1, peak protraction) and Γ(shape 3, peak retraction) shapes, first described in transilluminated juveniles (Drushel et al.,1997), are also observed in the MR images. Outlines must be drawn with caution, however, because some internal structures that are opaque to transillumination (thus casting a dark shadow) are not distinct from the`background' gray color in the MR images(Fig. 5D, dotted lines around elastic mouth tissue) or may appear white (indicating a high water content,e.g. the cartilaginous radular stalk). Shape space plots of buccal mass outlines from MR images may be different from those determined previously from transilluminated images simply as a result of the omission or inclusion of such differentially visible structures. In general, shape space analysis shows peak protraction (Fig. 3C,frame 3) to have greater ellipticity than transition/rest (frames 1 and 11)and peak retraction (frame 8) to have greater eccentricity than transition/rest or peak protraction, similar to transilluminated juveniles(compare Fig. 3C, which shows shape space analyses of transilluminated data, and Figs 8A, 10B, 12B, which show shape space analyses of MR data; see Fig. 3C for mathematical definitions of ellipticity and eccentricity parameters). As in the shape space plots of individual swallows from transilluminated juveniles (but not in the mean normalized swallow plot shown in Fig. 3C; see Fig. 7 of Drushel et al., 1997), MRI shape space plots show some hysteresis: the path taken through shape space from peak protraction back to rest is not the same as the path from rest to peak protraction (see especially Fig. 10B).

Internal structures

Mid-sagittal, high-temporal-resolution MR images of the buccal mass of adult A. californica during swallowing reveal significantly more internal anatomical details than corresponding latero-medial views of transilluminated juveniles (Fig. 5; Neustadter et al., 2002). The position, orientation and shape of the radula/odontophore within the buccal mass are clearly visible, as are the dorsal and ventral endpoints of the I1/I3 muscle (which establish the line of the lateral groove). The internal structure of the radula/odontophore can also be seen: the radular stalk, radular surface and 16 muscles are clearly differentiated. The mid-sagittal outline of the radula/odontophore itself changes shape during the feeding cycle. Qualitatively, during the progress from rest to peak protraction to peak retraction, the radula/odontophore outline changes from oval to round to rectangular. Such changes could only be inferred from the transilluminated buccal mass outlines.

I1/I3 muscle kinematics in transilluminated juveniles

The I1/I3 muscle complex changes shape as the radula/odontophore moves through it. These shape changes may be both passive (due to forces exerted by the radula/odontophore) and active (due to I1/I3 muscle contraction). A kinematic model cannot make any assumptions about the passive or active nature of the shape changes because no forces are represented. The buccal mass model used here assumes that the I1/I3 muscle is at rest unless it is being stretched by the presence of the radula/odontophore within its lumen. Thus,characterization of the actual I1/I3 muscle dimensions during swallowing in vivo provides important data with which to validate the performance of kinematic models of the radula/odontophore and the buccal mass.

The antero-posterior length of the dorsal I1/I3 muscle complex changes during the feeding cycle, as estimated from three consecutive swallows in a transilluminated juvenile (Fig. 6A). The minimum length occurs at peak protraction (t4/t1boundary), increasing to a maximum length at peak retraction (t1/t2boundary), after which there is no apparent change in length until the start of protraction (t3/t4 boundary). The maximum length is approximately 21% greater than the minimum length, which agrees well with the 17% length change observed in a dopamine-treated buccal mass in vitro(Drushel et al., 1998).

The medio-lateral width of the I1/I3 muscle complex also changes during the feeding cycle, and different regions change width at different rates,depending upon position along the antero-posterior axis(Fig. 6B). In the resting buccal mass, the I1/I3 complex is widest at the lateral groove and narrows to a minimum at the anterior end of the buccal mass. At all of six equally spaced reference `rings' along the antero-posterior axis, I1/I3 reaches a maximum width at peak protraction (t4/t1 boundary), and a minimum approximately 0.1 s (normalized) after peak retraction (t1/t2boundary), after which the widths increase to an intermediate level (during t3). The width change during protraction (t4) is greatest in the most anterior regions, with the sharpest increase approximately 0.2 s(normalized) before peak protraction. For example, the width of ring 6(anteriormost) increases by 0.6 mm (50%) during t4, whereas the width of ring 1 (posteriormost) increases by only 0.3 mm (10%). This 2:1 expansion ratio is also observed in high-temporal-resolution coronal sections of MR-imaged swallowing in vivo (see Fig. 9 of Neustadter et al., 2002a). During retraction (t1), the width of the more anterior regions decreases more quickly than that of the more posterior regions. This is consistent with the width changes expected by movement of the (relatively)spherical radula/odontophore through the I1/I3 complex during protraction and retraction.

Radular pitch angle in MR-imaged adults

In nine normalized swallows in an MR-imaged adult, the pitch angle of the radula (Fig. 1F) is constant(0° is horizontal) during protraction (t4), decreases to a minimum of approximately -70° (pitched up, towards the dorsal surface)during retraction (t1) and then returns to the initial value during transition (t3). This change in pitch angle is consistent with the change in radula/odontophore shape from round at peak protraction to rectangular at peak retraction and then back to oval during transition/rest(Fig. 5D-F).

The concept of `radular pitch' as a rigid body rotation is specific to the radula-centric kinematic model. However, the anatomical landmarks used to measure the pitch angle (i.e. a line from the postero-dorsal radular stalk to the anteriormost tip of the radular surface) are visible in all mid-sagittal MR images. Thus, the kinematics of the pitch angle should be reproduced by any model containing a representation of those landmarks irrespective of whether the pitch angle is a formal input. A smoothed version of Fig. 1F was used as an input for the radula-centric kinematic model.

Radula stalk rotation in MR-imaged adults

The rotation of the radular stalk (after removal of buccal mass rotation,buccal mass angle 0°) was measured in several different swallow sequences from mid-sagittal MR images of three different adults. In four normalized polyethylene tube swallows, the angle of the radular stalk(Fig. 1E) increases from 0°at rest to a maximum of approximately +90° midway through retraction(t1), falling to a plateau of 0° by the start of transition(t2/t3 boundary). In another polyethylene tube swallow (MRI sequence 7732-S3/15-38, see Fig. 10A),beginning from a rest position of -20° from vertical, the stalk rotates anteriorly to +80° at peak protraction. In a seaweed noodle swallow (MRI sequence 7725-S2/44-66, see Fig. 12A), beginning from a rest position of -10° (tilted posteriorly from vertical), the stalk rotates anteriorly during protraction(t4) and continues to reach a maximum of +90° during mid-retraction (t1) before rotating posteriorly to the rest value at the start of transition (t2/t3 boundary). The magnitudes of these rotations are similar.

The shape of the stalk angle curves suggests that rotation and translation of the radula/odontophore are independent movements (Neustadter et al., 2002). In Fig. 1E (see also Fig. 12A), at the peak of protraction as determined by the movement of the entire buccal mass(t4/t1 boundary), the radular stalk has completed only two-thirds of its final anterior rotation; this rotation is not completed until the buccal mass (and indeed the radula/odontophore) is already translating posteriorly during retraction. Stalk rotation continues for another 15° in the anterior direction even as the buccal mass is retracting posteriorly. In Fig. 10A, however, the peak anterior rotation corresponds to the peak of protraction (as determined by buccal mass movements, not internal movements of the radula/odontophore).

Radula-centric kinematic model

The radula-centric kinematic model of the radula/odontophore generates a range of continuous shapes by the roll, pitch and yaw of rigid-body radular halves according to parameters extracted from high-temporal-resolution mid-sagittal MR images of swallowing adults. The sequence of radula/odontophore shapes, when inserted into the buccal mass model, generates a continuous range of buccal mass shapes. Previous kinematic models of the buccal mass used fixed radula/odontophore shapes throughout the feeding cycle(Drushel et al., 1998). Model performance is evaluated by comparison with all available kinematic data.

Three-dimensional views

The radula-centric kinematic model yields realistic three-dimensional radula/odontophore shapes (Fig. 7) whose two-dimensional projections in lateral view resemble those of both transilluminated juveniles and MR-imaged adults(Fig. 5). Allowing for unmodeled dorsal buccal cavity tissue, the model accurately reproduces the ovoid, round and Γ shapes of rest/transition, peak protraction and peak retraction, respectively (Fig. 7A-C). The buccal mass shortens antero-posteriorly from rest to peak protraction and lengthens to a maximum at peak retraction.

As the radula/odontophore is protracted into the I1/I3 muscle complex, the rings expand and conform to the shape of the odontophore. Viewed dorsally, the I1/I3 complex changes from trapezoidal (rest, Fig. 7D) to rectangular (peak protraction, Fig. 7E). As the radula/odontophore retracts, the I1/I3 complex becomes trapezoidal again (peak retraction, Fig. 7F) and is narrower medio-laterally than at rest. This is due to shape changes of the radula/odontophore: at rest (and for the first half of protraction), the radular halves are open (roll and yaw, Fig. 1G), and the location of the lateral groove and hinge is such that the posterior I1/I3 rings must stretch over the wide radula/odontophore. At peak retraction, the I1/I3 rings are completely closed upon themselves in the mid-sagittal plane (Fig. 7I). In addition, because the radular halves are rigid bodies whose basic shape was defined from a fully closed radula/odontophore, their opening creates a blunt shape to which the I1/I3 cannot conform without gaps(Fig. 7G). Even when closed,the radula/odontophore is too wide for the I1/I3 muscle model at peak protraction when the large dorsal surface of the radula/odontophore is rotated approximately 90° anteriorly into the jaws; additional gaps then appear between the I1/I3 rings (Fig. 7H).

While the endpoints of the I7 muscles are anatomically consistent with the placement of the radular halves and radular stalk (based upon MR images), the cylindrical representation in inaccurate because of the unmodeled volume at the endpoints. Maintaining an isovolumetric constraint only for the central portion of the I7 muscles makes them too thick at peak protraction(Fig. 7B). Finally, as a result of over-simplification of its shape, the radular stalk cylinder overlaps with the I7 muscles during model frames around peak retraction(Fig. 7C). (For simplicity, the radula-centric model does not attempt to make the I7 muscles conform to any underlying structures.)

Quantitative results

The shape space plot of the radula-centric model(Fig. 8B) is a continuous closed path that occupies the same region of shape space as the plots of swallows in transilluminated juveniles(Fig. 3C). It also has rough similarities to that of MRI sequence 5385-S1/116-133, the swallowing sequence upon which its I1/I3 placement parameters were based(Fig. 8A). The radula-centric model has a clear hysteresis loop from peak protraction to peak retraction(frames 29-37), similar to the individual swallows in transilluminated juveniles (Drushel et al.,1997); however, the MRI shape space plot has no clear hysteresis. The large positive ellipticity value (i.e. the shape is taller dorso-ventrally than long antero-posteriorly) and small positive eccentricity (i.e. the shape is skewed more posteriorly than anteriorly) for peak retraction in the MRI data (frame 128, Fig. 8A) are due to the presence of elastic jaw tissue, clearly different from the I1/I3 muscle, which was not included in the traced outline. Ring 5 of the modeled I1/I3 muscle complex accounts for this non-I1/I3 tissue, shifting the entire shape space plot to the left.

The lengths of the modeled I2, I7 and I1/I3 muscles show changes during the feeding cycle consistent with the movements of the radula/odontophore and buccal mass. The I2 muscle (Fig. 8C) shortens to 60% of rest length during protraction(t4) and lengthens to 158% of rest length during retraction(t1); the overall length change is approximately 270% of the minimum. The length of the I7 muscle (Fig. 8C) mostly follows the change in radular pitch angle(Fig. 1G): constant with a small decrease to 80% of rest length before the peak of protraction(t4/t1 boundary), then increasing to 320% of rest length until half-way through retraction (t1), and finally a return to the rest length during t2 and t3 (peak retraction through transition); the overall length change is approximately 400% of the minimum. The antero-posterior length of the I1/I3 muscle complex(Fig. 8D) decreases to 90% of rest length approximately two-thirds of the way through protraction(t4), then increases to 110% of rest length during retraction(t1); the overall length change is approximately 125% of the minimum.

The individual I1/I3 rings (Fig. 8E) change their mediolateral width to conform to the radula/odontophore shape as it is moved through them. The posterior rings (1 and 2) are wide at rest since they are close to the lateral groove and overlap the radula/odontophore, whereas the anterior rings (3-5) are closed and narrow. The anterior rings expand very sharply (frames 6-14) as the open,blunt-faced radula/odontophore protracts through them; the posterior rings actually shrink slightly as narrower stalk regions of the radula/odontophore are rotated anteriorly. A similar sharp decline in width is seen in all rings during mid-retraction (t1), at which point (frames 41-52) the radula/odontophore is completely posterior to all the rings and is undergoing internal movements that do not impinge upon the rings. During the transition period (t3), the radula/odontophore moves anteriorly to its rest position and the I1/I3 rings open to accommodate it. Ring 4 experiences the largest changes in medio-lateral width, approximately 285% of its minimum.

The radula-centric model of the radula/odontophore duplicates the overall shape changes of the buccal mass and relates radular opening and closure to radula/odontophore rotation within the buccal mass. The fact that rigid bodies are used to represent the radular halves, however, leads to sudden,non-physiological width changes in the modeled I1/I3 muscle complex. Radular opening and closure are likely to be more complicated than a set of rigid-body rotations. Unmodeled structures are likely to contribute significantly to the overall radula/odontophore shape. We thus turned to an odontophore-centric model in an attempt to construct radula/odontophores whose shapes were more explicitly constrained by the non-analytical shapes seen in high-temporal-resolution mid-sagittal MR images. Swallows of both polyethylene tubes and seaweed noodles were modeled because both methods of feeding seemed successful in limiting out-of-plane animal movements (necessary to obtain precise mid-sagittal images) while eliciting vigorous feeding responses.

Odontophore-centric kinematic model, MRI sequence 7732-S3/15-38(polyethylene tube swallow)

Three-dimensional views

The odontophore-centric kinematic model was used to generate a three-dimensional representation of a parallax-free MR-imaged swallowing sequence, 7732-S3/15-38 (Fig. 9). This corresponds to swallow 3 of Neustadter et al. (2002). The animal in this sequence was swallowing a polyethylene tube after arousal with aqueous seaweed extract. Unlike the radula-centric model, no averaged or composite data from multiple swallows were used, only measurements taken from this particular sequence. Only four I1/I3 rings were needed to represent the I1/I3 muscle complex (compared with five used for the radula-centric model). The model radula/odontophore outlines are very similar to those observed in the corresponding MR images (compare Fig. 9A-C and Fig. 9G-I). Unlike the radula-centric model, the rest and peak-protraction radula/odontophore models are not excessively wide medio-laterally (compare Fig. 7D,E and Fig. 9J,K). Furthermore, the mediolateral width of the overall buccal mass is correctly widest at peak protraction(Fig. 9K) in accord with dorso-ventral video recordings from transilluminated juveniles(Drushel et al., 1997). The radula/odontophore is narrowest at peak retraction(Fig. 9L). The Γ shape(Fig. 9I) is not as pronounced as in the radula-centric model (Fig. 7C) because the 7732-S3/15-38 sequence has a weak retraction(Fig. 9C; compare Fig. 5F and Fig. 11C) and the model run attempts to reproduce the MR images as exactly as possible. Nonetheless, the isovolumetric constraint upon shape changes causes the elongated radula/odontophore of peak retraction to have a narrow medio-lateral width(volume is redistributed dorsally and ventrally).

The I1/I3 muscle complex is adequately represented during the first part of protraction (t4), the last part of retraction (t1) and the remainder of the feeding cycle through transition (t2 and t3). Around peak protraction, however, the model begins to turn itself inside out: the dorsal I1/I3 muscle rings are too far posterior, even though the ventral I1/I3 rings are correctly attached(Fig. 9H,K). This results in the radula/odontophore protruding through the anteriormost ring, as if it were a model of a full bite rather than a swallow. In the actual MR image, the radula/odontophore is clearly posterior to the opening of the jaws(Fig. 9B). The region corresponding to the modeled first I1/I3 ring seems to be stretched posteriorly at peak protraction (Fig. 9B, dotted line), leaving the region corresponding to the second ring in nearly the same position (relative to the opening of the jaws) that it occupied at rest (Fig. 9A). At peak retraction (Fig. 9C), the ventral I1/I3 region no longer appears to be stretched, and the actual I1/I3 muscles again correspond closely to the modeled rings. At no point, however,do any of the I1/I3 rings pull apart from each other as in the radula-centric model; the odontophore-centric radula/odontophore (by construction) lacks blunt, sharply flattened surfaces that would cause the rings suddenly to increase in medio-lateral width.

The I7 muscle changes length as its endpoints change on the surface of the radula/odontophore. Because the volume of the attachment points is explicitly represented (by cones), together with that of the cylindrical belly, the I7 muscle diameter changes are much less pronounced than in the radula-centric model representation (Fig. 9G-L). In addition, since the anterior endpoints are not attached to rolling or yawing radular halves (compare Fig. 1C), the I7 muscles always run parallel to the mid-sagittal plane. The anatomically correct radular stalk(reconstructed from high-spatial-resolution MR images of an anesthetized buccal mass) is narrower in diameter dorsally than ventrally, so that the modeled I7 muscle is able to (effectively) pitch posteriorly without colliding with the modeled stalk (compare Fig. 7C and Fig. 9I).

Quantitative results

The shape space plots of both the original MR-imaged swallow(Fig. 10B) and the odontophore-centric model (Fig. 10C) occupy a larger region of shape space than those of transilluminated juveniles (Fig. 3C). In both plots, the points around peak protraction have large positive ellipticity. In the case of the MR images, this is due to omission of anterior elastic tissue from the buccal mass outline. The odontophorecentric model plot, however, has even higher ellipticity because the I1/I3 muscle rings are too far posterior (Fig. 9H) as a result of stretching of the ventral I1/I3 muscle area seen in the MR images (Fig. 9B,dotted line). The foreshortened I1/I3 ring stack also affects the eccentricity parameter, causing the model buccal masses to be skewed anteriorly (negative eccentricity, frames 19-22). Rest (frames 15 and 38) and peak retraction(frame 35), however, are comparable between MR images and model(Fig. 10B,C).

Odontophore-centric model representations of the I2, I7 and I1/I3 muscles show length changes that are qualitatively similar to those in the radula-centric model, but quantitatively less extreme. The I2 muscle(Fig. 10D) decreases to 50% of rest length during protraction (t4), then increases to 110% of rest length by the peak of retraction (t1/t2 boundary), after which it returns to the rest value; the overall length change is approximately 210% of the minumum. After an initial increase of 50% from rest length, the I7 muscle(Fig. 10D) remains at constant length throughout protraction (t4). During retraction (t1),I7 increases to 250% of rest length, reaching a maximum at peak retraction(t1/t2 boundary), then returning to the rest value. The antero-posterior length of the I1/I3 muscle complex(Fig. 10E) decreases to 85% of rest length during protraction (t4), then increases to 110% of rest length during retraction (t1) and through the loss of the Γshape (t2), and finally decreases to the rest length; the overall length change is approximately 130% of the minimum. The I1/I3 muscle rings change width more gradually than in the radula-centric model, but there are still some rapid transitions (Fig. 10F; compare Fig. 8E). The posteriormost rings (1 and 2) open abruptly as the radula/odontophore begins the anterior rotation of protraction (at rest, the radula/odontophore is posterior to all four rings, so all are completely closed), but the anterior rings (3 and 4) open more gradually. The rings close rapidly in an antero-posterior wave during the second half of retraction(t1). Ring 3 experiences the greatest length change, an approximately 210% increase from the minimum (rest) value.

Odontophore-centric kinematic model, MRI sequence 7725-S2/44-66(seaweed noodle swallow)

Three-dimensional views

The odontophore-centric kinematic model was used to generate a three-dimensional representation of a second parallax-free MR-imaged swallowing sequence, 7725-S2/44-66 (Fig. 11). This corresponds to swallow 2 of (Neustadter et al., 2002). The adult in this sequence was swallowing a seaweed noodle, which unrolled continuously from a spool. Unlike the 7732-S3/15-38 sequence, this second swallow has a strong retraction and a pronounced Γ shape at peak retraction (Fig. 11C), so the corresponding odontophore-centric model frame(Fig. 11I) more closely resembles the radula-centric model frame(Fig. 7C). Because the dorsal and especially ventral outlines of the I1/I3 muscle complex are so tortuous(Fig. 11A, curved lines), six rings of different increasing and then decreasing (along the antero-posterior axis) rest diameters were used to attempt to model the outlines accurately.(The dorsal and ventral limit lines for the I1/I3 muscle rings are constrained to be straight lines, except where the rings would be in contact with the radula/odontophore; see Fig. 2Dof Drushel et al., 1998.)

In this model sequence, the medio-lateral widths of both the buccal mass and the radula/odontophore decrease over the feeding cycle(Fig. 11J-L). At rest, the radula/odontophore is much wider than in the corresponding model frame of the 7732-S3/15-38 sequence (Fig. 9J), even though the odontophore volumes are the same between the two model runs. While the rest radula/odontophore is nearly as wide as that in the radula-centric model (Fig. 7D), it is much narrower in peak protraction (compare Fig. 7E and Fig. 11K), so that the I1/I3 muscle rings do not have gaps due to excessive stretching (antero-posterior views not shown). Both the buccal mass and the radula/odontophore are narrowest at the peak of retraction as volume is redistributed ventrally to accommodate the protrusion of the stalk in the Γ shape.

The model successfully matches the tortuous outlines of the I1/I3 muscle complex at rest (Fig. 11G),but becomes less accurate as protraction proceeds until, at peak protraction,the buccal mass hyper-protracts (Fig. 11H). The ventral I1/I3 muscle corresponding to the first model ring is stretched even further than in the 7732-S3/15-38 sequence (compare Fig. 9B and Fig. 11B, dotted lines). To maintain the hinge attachment of the ventral ring 1 with the radula/odontophore, the lateral groove is placed too far posteriorly and the radula/odontophore is protracted far through the anteriormost ring, as if in a strong bite. The MR image clearly shows that the radula/odontophore is no further forward than the anteriormost region of the I1/I3 muscle(Fig. 11B). The small ring 6,used to match the anteriormost outline of the I1/I3 muscles at rest, has been stretched so far (through the isovolumetric constraint) that it is largely inside the lumen of ring 5. Postero-dorsally, rings 1 and 2 are closed upon what looks like empty space; this is actually the dorsal limit line extending posteriorly to the odontophore surface. The model is simply obeying its internal ring placement rules, but the rings are in an anatomically inappropriate area, so the results are inaccurate. The latero-medial view of peak retraction (Fig. 11I) is qualitatively accurate, yet the anteriormost rings have an unnatural posterior tilt as they begin to nest within each other's lumina.

The appearance of the I7 muscle (Fig. 11G-I) is similar to that in the odontophore-centric model of the 7732-S3/15-38 sequence (Fig. 9G-I). Though represented isovolumetrically as its endpoints change, I7 has small changes in diameter throughout its range of length changes. The modeled I7 muscle never collides with the radular stalk.

Quantitative results

Shape space plots of both the original MR-imaged swallow(Fig. 12B) and the odontophore-centric model (Fig. 12C) are spread out over a large region of shape space compared with transilluminated juveniles (Fig. 3C). The MRI shape space plot is very similar to a transilluminated plot, with the large positive ellipticity (i.e. taller dorso-ventrally than long antero-posteriorly) of protraction frames (48-53)being due to the lack of non-I1/I3 elastic tissue being included in the buccal mass outline. The model shape space plot has protraction frames that have even greater positive ellipticity because the I1/I3 ring stack is too far posterior, causing the buccal mass to be too short antero-posteriorly. This also causes model frames entering peak protraction (54-59) to have negative eccentricity (i.e. skewed anteriorly) because the protruding radula/odontophore of retraction is too close to the (too short) anterior end of the modeled buccal mass. Finally, transition and rest frames (44 and 45, 65 and 66) have too large a positive eccentricity (i.e. skewed posteriorly) and too negative an ellipticity (i.e. elongated along the antero-posterior axis),because of the extended six-ring I1/I3 muscle stack.

The I2, I7 and I1/I3 muscle lengths in the odontophore-centric model of the MRI swallow sequence 7725-S2/44-66 show changes that are similar in magnitude to those of the 7732-S3/15-38 sequence. The I2 muscle(Fig. 12D) shortens to 53 % of rest length during protraction (t4), lengthens to a maximum of 120 %of rest length at peak retraction (t1/t2 boundary) and then returns to rest length; the overall length change is 220 % of the minimum. The I7 muscle (Fig. 12D) remains close to its rest length during protraction (t4), and then lengthens to 190 % of rest length by mid-retraction (t1), after which its length remains constant until peak retraction (t1/t2 boundary),finally returning to the rest value by the end of transition (t3). The antero-posterior length of the I1/I3 muscle(Fig. 12E) decreases to 78 %of rest length until just before peak protraction (t4/t1 boundary);it then rises to a plateau of 83 % of rest length through most of retraction(t1) and increases to a maximum of 107 % of rest at the start of transition (t2/t3 boundary) before a two-frame decline back to the rest length. The total range of length change is 138 % of the minimum. The I1/I3 rings undergo some abrupt changes in medio-lateral width during protraction (t4). Because the I1/I3 ring stack is too far posterior,the posteriormost rings (1 and 2) actually reach their maximum width early in protraction and then become narrower; they are posterior to the thickest part of the radula/odontophore and are closing down upon the stalk as the radula/odontophore rotates anteriorly. At peak protraction (t4/t1boundary), the widest part of the buccal mass is actually due to rings 3-5. The ring widths decline in two steps during retraction (t1): an initial slow decrease for most of the interval, followed by a faster three-frame decline immediately before peak retraction (t1/t2)boundary). Kinematically, the I1/I3 rings are closing as designed but, by the time of peak protraction, because of the unmodeled posterior stretching of the ventral portion of ring 1, they are incorrectly positioned anatomically. Ring 5 shows the greatest change in length, an approximately 230% increase from the minimum (rest) value.

We have created two kinematic models of the radula/odontophore of A. californica to characterize further the movements of structures inside the buccal mass during the feeding cycle. Both models produce a continuous range of shape changes in the radula/odontophore, but the models are fundamentally different in construction. The radula-centric model treats the radular halves as rigid bodies and hypothesizes that the overall shape of the radula/odontophore can be explained by simple pitch, yaw and roll rotations of these bodies relative to a fixed radular stalk. In contrast, the odontophore-centric model creates a globally convex solid representation of the radula/odontophore directly, which must then constrain the positions and shapes of internal structures. Both radula/odontophore models are then placed into an existing kinematic model of the buccal mass(Drushel et al., 1998) to generate three-dimensional representations. Hightemporal-resolution mid-sagittal MR images of swallowing adults in vivo are used to provide accurate shape and position parameter inputs for the models free from in vitro artifacts. The MR images allow some structures inside the buccal mass to be seen directly, whereas previous non-invasive visualization techniques (transillumination of semitransparent juveniles) showed changes only in the outline of the entire buccal mass. The three-dimensional models are a synthesis of all available kinematic data and make predictions about the shape and position of internal structures that are either invisible or only partially visible (depending upon technique and imaging plane).

Comparison of models: differences in radula/odontophore representations

Here, we describe differences between the radula-centric and odontophore-centric models of the radula/odontophore with regard to the radula/odontophore itself (i.e. its own three-dimensional shape and internal structures). The effects of these differences upon the overall buccal mass kinematic model are considered in a separate section of the Discussion (see below).

Radula/odontophore outlines in the mid-sagittal plane

Both the radula-centric and odontophore-centric models are capable of reproducing the shape changes observed in high-temporal-resolution mid-sagittal MR images during the feeding cycle (Figs 5D-F, 9A-C, 11A-C). A detailed kinematic analysis of the mid-sagittal MR images is presented elsewhere(Neustadter et al., 2002a). Briefly, the radula/odontophore is oval at rest, more rounded at peak protraction and elongated parallel to the radular stalk axis at peak retraction. These MR images confirm the prediction of previous kinematic modeling studies in which single in-vitro-derived radula/odontophore shapes were unable to reproduce the overall buccal mass outlines observed in transilluminated juveniles over a probable 90° range of radula/odontophore rotation (Drushel et al.,1998).

Medio-lateral width and shape

The radula-centric and odontophore-centric models differ significantly in their prediction of the medio-lateral width of the radula/odontophore. Although it is anatomically reasonable to represent radular halves that rotate open or closed (see Fig. 10 of Drushel et al., 1998), treating them as rigid bodies leads to large overestimates of the medio-lateral width in the resulting radula/odontophore model if the radular halves are open. The odontophore-centric model is designed to be smoothly convex in all dimensions,and it changes mediolateral width only to maintain a constant radula/odontophore volume as the mid-sagittal outline changes shape.

The exact three-dimensional medio-lateral shape of the radula/odontophore is not known at high temporal resolution. In the radula-centric model, the 14 muscles are not explicitly modeled; instead, the outer margin of the radular halves is assumed to represent the widest extent of the radula/odontophore,and the volume is filled in with ellipse quadrants along the antero-posterior axis (Fig. 1D). Similarly, in the absence of further constraints, the four out-of-mid-sagittal-plane superellipse arcs of the odontophore-centric model are assumed to be simple ellipses (parameter n=1.0). The mediolateral width achieved by the constant-volume calculator might be significantly different were the n parameters of these four superellipse arcs changed to give flatter or more convex shapes. Without an analysis of the three-dimensional kinematics of the radula/odontophore, insufficient data exist to derive functions to change the four out-of-plane n parameters with time or rotational position.

By definition, medio-lateral width is directly related to radular opening in the radula-centric model: the greater the yaw angle, the wider the radula/odontophore. However, the yaw and roll angle parameters were chosen to reproduce a swallow, not a bite (Fig. 1G), so the radular halves are not open at peak protraction, and the radula/odontophore is narrower medio-laterally at peak protraction and peak retraction than at rest (Fig. 7D-F). It is notable that the odontophore-centric model also reproduces these width changes, having no structures that explicitly open or close in the medio-lateral direction, using only a constant-volume constraint as the mid-sagittal outline changes shape. In the odontophore-centric model of MRI sequence 7732-S3/15-38 (polyethylene tube swallow), the radula-odontophore is approximately the same width from rest to peak protraction and narrows only slightly at peak retraction (Fig. 9J-L). In the odontophore-centric model of MRI sequence 7725-S2/44-66 (seaweed noodle swallow), the radula/odontophore is widest at rest, and both peak protraction and peak retraction are approximately the same width (Fig. 11J-L). In anesthetized buccal masses in vitro, minimally invasive dissection shows that the radula/odontophore is positioned within the buccal mass much as shown in Fig. 7D, with the halves open posteriorly by approximately 5° (data not shown). A directly illuminated juvenile whose radula was stained with the vital dye Fast Green had a slightly open radula at rest (see Fig. 8 of Drushel et al., 1998). From the available evidence, we hypothesize that, during swallowing, the radula is open from the loss of the Γ shape (t2/t3 boundary) to some point in late protraction (t4). Analysis of coronal MR images of swallows supports this hypothesis (Neustadter et al., 2002b).

I7 muscle length

Both kinematic models of the radula/odontophore predict the length of the I7 muscle throughout the feeding cycle. The length of muscle I7 quadruples in the radula-centric model (Fig. 8C), but only doubles in the odontophore-centric model (Figs 10D, 12D). Minimum length occurs near the start of protraction (t3/t4 boundary), and maximum length occurs in the latter half of retraction (t1). These two predictions differ because of differences in the representation of the I7 muscle. In the radula-centric model, the dorsal attachment of the I7 muscle is a point affixed to a radular `skirt' (a band of radular membrane ventral to the junction of radular halves, which is not explicitly represented as an isosurface yet which moves with the represented volume of each radular half;see Fig. 1A-C). In the odontophore-centric model, the dorsal endpoint is chosen by following the trajectory of the I7 muscle antero-dorsally through the dorsalmost extent of the I6 muscle to intersect the radular surface at the anterior tip of the radula (Fig. 2B). Since the radular stalk (and hence the antero-ventral attachment of the I7 muscle) is a fixed reference point in both models, I7 muscle length changes are accomplished in different ways. In the radula-centric model, any rotation of the radular halves (but especially pitch) will move the dorsal endpoint; in the odontophore-centric model, the dorsal endpoint is determined by direct inspection of mid-sagittal MR images. The large percentage change in I7 muscle lengths in the radula-centric model is probably due to error in specifying I7's dorsal endpoint because of over-rigidity of the shape of the radular half during pitch. The odontophore-centric model probably also slightly overestimates the I7 length change, since the I7 muscles are actually parasagittal, and the true dorsal endpoint is posterior or ventral to the mid-sagittal outline seen in the MR images (as a result of the three-dimensional curvature of the radula/odontophore). The I7 muscle has approximately linear length/tension properties over a 225% change in length(Evans et al., 1996), so the approximately 220% length change of I7 predicted by the odontophore-centric model is not outside the physiological range.

Radular stalk

The tapered, anatomically correct radular stalk used in the odontophore-centric model avoids problems of kinematic interference with the I7 muscles compared with the simple cylinder used in the radula-centric model(Fig. 7C,I). Since neither model implements the ability of muscles to conform dynamically to other solid objects, this does not adversely affect the estimate of the I7 muscle lengths. Indeed, the I7 muscles would be even longer in the radula-centric model. No further insights were to be gained by re-running the radula-centric model with the improved radular stalk model.

Comparison of models: differences in buccal mass model representations

Here, we discuss the effect of the two different radula/odontophore models upon the output of the buccal mass model. This includes the dimensions of the I1/I3 and I2 muscles and shape space analysis of mid-sagittal views of the buccal mass. Since the underlying shape of the radula/odontophore is a causal input to the buccal mass model, the radula-centric and odontophore-centric models may affect the modeled I1/I3 and I2 muscles differently, and provide additional validation tests.

I1/I3 muscle medio-lateral width

As noted above, both the radula-centric and odontophore-centric models of the radula/odontophore result in I1/I3 muscle width changes that are larger and more abrupt than those observed in transilluminated juveniles in vivo (compare Fig. 6B with Figs 8E, 10F, 12F). In the radula-centric model, the width changes actually cause the modeled I1/I3 muscle rings to pull apart from one another (Fig. 7G,H, arrows). The I1/I3 muscle rings are implemented kinematically, so they must conform to the shape of the underlying radula/odontophore model. Moreover, the I1/I3 muscle rings are kinematically linked only at the dorsal and ventral midpoints, not along their entire circumference. Thus, extreme changes in medio-lateral width of the radula/odontophore can lead to failures of the I1/I3 muscle model.

While it is clear that the I4 muscles and cartilaginous bolsters beneath each half of the radula must be capable of some medio-lateral movement during radular opening and closure, the structures are soft and deformable, and the enclosing I1/I3 muscles probably smooth out the radula/odontophore shape. A full kinetic model of the buccal mass, in which muscles generate actual forces, would be expected to have this property. A kinematic model of the radula/odontophore, however, must `anticipate' the deformations due to in vivo I1/I3 muscle forces for the kinematically placed I1/I3 model rings to assume the correct shapes. In this respect, the current I1/I3 muscle model can only be as correct as the three-dimensional radula/odontophore shapes that are moved through it.

I1/I3 muscle antero-posterior length

Both radula/odontophore models result in I1/I3 muscle antero-posterior length changes that are similar in timing and magnitude to those observed in transilluminated juveniles (compare Fig. 6A with Figs 10E, 12E). The number of I1/I3 rings used to model a particular sequence can affect the antero-posterior length change. In the four-ring model of the 7732-S3/15-38 MRI sequence, the I1/I3 antero-posterior length range was 110% of the minimum length, whereas in the six-ring model of the 7725-S2/44-66 sequence, the range was 138%. Length changes were within physiological ranges despite the fact that the lateral groove (i.e. the position of the first I1/I3 muscle ring) was too posterior in both odontophore-centric model sequences.

I2 muscle length

In both models, the 12 muscle is shortest at peak protraction(t4/t1 boundary) and longest at peak retraction (t1/t2boundary). The total length change (per cent of minimum) is greater in the radula-centric model (270 %) than in the odontophore-centric model (210 % and 220 %). These are greater than the 150 % length change estimated from transilluminated juveniles (Drushel et al.,1998), the 160 % length change measured from mid-sagittal MR images of swallowing adults (Neustadter et al., 2002a) and the 160 % length change predicted by the previous,fixed-radula/odontophore-shape kinematic model(Drushel et al., 1998). Currently, all estimates of I2 muscle length in vivo are overestimates since measurements assume that the I2 muscle runs along the mid-sagittal outline of the posterior buccal mass, between the dorsal and ventral I1/I3 muscles. Part of the I2 muscle is actually parasagittal, with paired dorsal halves running on either side of the esophagus, and its true length depends upon the three-dimensional shape of the radula/odontophore out of the mid-sagittal plane. In addition, model-derived predictions of the I2 muscle length depend upon the plane of the posteriormost I1/I3 muscle ring. Especially in the odontophore-centric model runs presented here, the dorsal I1/I3 is positioned too far posteriorly. Thus, the estimated I2 muscle lengths are probably too short in peak protraction and too long in peak retraction,leading to a large total length change.

Shape space analysis

Shape space plots (of ellipticity versus eccentricity measurements, see Fig. 3 for definitions) are roughly comparable between a given MRI sequence and its kinematic model representation, but less comparable between different MRI sequences or with the `standard' shape space plot derived from transilluminated juveniles (Drushel et al.,1997). Shape space was devised to describe quantitatively complex changes in the outline of the buccal mass and was used later as a criterion for the validation of kinematic models(Drushel et al., 1998). The significantly greater detail provided by high-temporal-resolution MRI,especially of heretofore unobservable structures inside the buccal mass,reduces the utility of shape space analysis. The buccal mass undergoes more complex shape changes than could be observed previously. Approximation of the mid-sagittal buccal mass outline as four ellipse quadrants cannot adequately capture these details and has proved a constraint to which the MRI data cannot be fitted without significant error. We therefore propose that higher-resolution shape-matching techniques be developed to evaluate the performance of kinematic models relative to MRI data(Neustadter et al., 2002b) to supersede our shape space analysis.

Comparison of feeding stimuli

We do not believe that the small differences observed between the MR-imaged swallows of polyethylene tubes (Figs 9A-C, 10A,B) and seaweed noodles(Figs 11A-C, 12A,B) are significantly different for the purposes of evaluating the performance of the odontophore-centric model (Figs 9G-L and 10C-F compared with Figs 11G-L and 12C-F). Both stimuli elicited continuous, multi-swallow feeding responses and helped to maintain proper alignment of the animals within the visualization plane of the MRI apparatus. Small real differences in handling of the two food stimuli may exist, but are below the ability of the present study to resolve. Since polyethylene tubes can also be used to elicit rejection responses(Morton and Chiel, 1993a),they may be ideal stimuli for detailed MRI-based studies of rejection kinematics in vivo, leading to improved kinematic (and kinetic)models of feeding in A. californica.

Implications for future kinematic models

Radula/odontophore model

The current radula-centric model, while producing realistic motions, is too simple in its internal construction to reproduce the full three-dimensional shape of the radula/odontophore. Components important for generating the correct medio-lateral shape are missing (namely, the 14 muscle and bolsters; Starmühlner, 1956). The existing odontophore-centric model might be able to generate the appropriate shapes if functions that relate the change of the four out-of-plane superellipse n parameters could be determined experimentally. A non-superellipse-based odontophore-centric model, incorporating detailed quantitative kinematic analysis of mid-sagittal MR images(Neustadter et al., 2002a) and in vitro preparations, is under development(Neustadter et al., 2002b). We believe that a sufficiently accurate odontophore-centric model will provide the constraints necessary to construct a revised `inside-out'radula/odontophore model, which is built up from the necessary internal components. Minimally, such a `component-centric' model will contain the radular stalk, the I4 muscle, the bolsters, the I6 and the I5 muscles.

I1/I3 muscle model

The I1/I3 ring model uses straight lines to limit the dorso-ventral closure of each ring; this kinematic representation of the dorso-ventral stiffness of the jaw cartilages prevents symmetrical, sphincter-like closure of the I1/I3 muscles (Drushel et al., 1998). A straight line was a reasonable inference from transilluminated images(Fig. 5A-C), but midsagittal MR images show that the dorsal and ventral outlines of the I1/I3 muscles can actually be curved, ranging from mostly straight(Fig. 5D-F) to highly tortuous(Fig. 11A). The I1/I3 ring model could be revised to allow curved limit lines derived from polynomial fits to actual mid-sagittal MR image outlines(Neustadter et al.,2002b).

The excessively posterior placement of the I1/I3 muscle rings at peak protraction in the odontophore-centric model leads to a non-physiological appearance of the dorsal half of the I1/I3 complex (Figs 9B,H, 11B,H). The kinematic rule that the plane of the posteriormost I1/I3 muscle ring (i.e. the lateral groove) should be perpendicular to the mouth—esophagus (buccal mass)axis of the buccal mass throughout the feeding cycle (deduced from images of transilluminated juveniles) is probably incorrect. In high-temporal-resolution MR images of peak protraction, the apparent posterior stretching of the ventral I1/I3 muscle at the hinge region causes the lateral groove to tilt approximately 30° anteriorly from the perpendicular (Figs 9G, 11G). Constraining the anteriormost I1/I3 muscle ring (i.e. the jaw line) to be perpendicular to the buccal mass axis may be more accurate (compare Fig. 11C and 11I).

I2 muscle model

An improved kinematic model of the I2 muscle requires both an accurate three-dimensional shape for the underlying radula/odontophore and accurate placement of its dorsal and ventral endpoints along the lateral groove. The known limitations of the current I2 muscle model do not invalidate the basic prediction that I2 is shortest at peak protraction and longest at peak retraction.

Functional implications of kinematic models

The complex anatomy of the molluscan buccal mass makes it difficult to assign functional roles to individual muscles a priori. The buccal mass consists only of soft tissues (muscle and cartilage); there are no bones or other rigid anchor points internally or externally. Unlike A. californica, in which a neutral `intrinsic' or `extrinsic' muscle nomenclature is used (Howells,1942), other molluscan systems have a functional nomenclature for buccal mass muscles (Carriker,1946; Smith,1990). Care must be taken when ascribing functions to muscles in the molluscan buccal mass (Murphy,2001). Electromyographic recordings from individual muscles can show patterns of activity (Kater,1974; Rose and Benjamin,1979), and stimulation of individual muscles by fiber shock(Bulloch and Dorsett, 1979) or nerve shock (Evans et al.,1996) may suggest muscle actions, but the resulting movements are dependent upon the balance of all the forces (active and passive) in the system, and this is highly dependent upon the instantaneous geometry of all the components. Non-invasive techniques of observation, such as direct illumination (Rose, 1971; Rose and Benjamin, 1979),transillumination (Drushel et al.,1997), sonomicrometry(Orekhova et al., 2001) and MR imaging (Neustadter et al.,2002a) are necessary to acquire accurate kinematic data. Kinematic models are a useful way to integrate kinematic data with test hypotheses about the functional interconnections of the structures within the buccal mass.

It is possible that some muscles that have been identified on the basis of electromyographic or lesion studies as protractors(Hurwitz et al., 1996),retractors (Morton and Chiel,1993b), radula openers (Evans et al., 1996) or radula closers(Orekhova et al., 2001) may act not through pure strength, but rather through delicate coordination with other components. For example, lesions of the I2 muscle result in greatly weakened protraction of the radula/odontophore, such that bites (i.e. protraction of the radular surface completely through the jaws) are impossible even in fully aroused, hungry animals(Hurwitz et al., 1996). The I2 muscle, however, is very thin, and the thick I1/I3 muscle complex seems very powerful. Could I2 muscle contractions alone overpower a fully contracting I1/I3 muscle complex? Perhaps the function of the I2 muscle is to provide an initial protraction, enough to position the radula/odontophore precisely such that contractions of other, more powerful, muscles (I1/I3 or I6) actually accomplish the main protraction movement. We hypothesize that the functions of the intrinsic muscles of the A. californica buccal mass are highly dependent upon the local context in which they find themselves at any point in the feeding cycle and that there are few, if any, muscles with a single action.

On the basis of electrophysiological studies, the generalized molluscan feeding cycle is hypothesized to consist of three phases: `protraction',`retraction' and `hyper-retraction'(Murphy, 2001). The terms`protraction' and `retraction' are problematic, however, because they have been employed to describe movements both of the entire buccal mass (caused by both intrinsic and extrinsic muscles) and of the radula/odontophore. We have previously stated operational definitions of `protraction' and `retraction' to be any anterior or posterior movements, respectively, of the radula/odontophore (Drushel et al.,1997), although other workers restrict usage to subsets of such movements, using both functional and electrophysiological criteria(Murphy, 2001). Careful attention to these nomenclature differences is necessary to avoid misinterpretation of published data. In A. californica, normal swallowing consists of smooth, continuous movements of the radula/odontophore. Measurements of the change in angle of the radular stalk relative to the buccal mass axis in MR images reveal only a smooth, continuous retraction movement (Figs 1E, 10A, 12A). It is possible, however,that the movements of the radular stalk do not always reflect movements of the rest of the odontophore (Neustadter et al., 2002a). During tearing behavior in transilluminated juvenile A. californica, there is a vigorous posterior radula/odontophore rotation with a sharp posterior `snap' as a piece of food breaks free,followed by an exaggerated ventral protrusion of the radular stalk (R. F. Drushel and H. J. Chiel, unpublished observations; see also Hurwitz and Susswein, 1992),which might be considered a functional hyper-retraction. Mid-sagittal MR images of A. californica at peak retraction during swallowing (Figs 5F, 9C, 11C) show the same strong stalk protrusion as the hyper-retracted position of Helisoma trivolvis (Fig. 1,position 5 in Murphy, 2001). Perhaps, as in Helisoma trivolvis(Arnett, 1996; A. D. Murphy,personal communication), the `retraction' and `hyper-retraction' neural phases in A. californica correspond (respectively) to the active posterior rotation of the peak-protraction radula/odontophore to its rest position and the subsequent active protrusion of the radular stalk, with final posterior rotation of the radular surface to the esophageal opening. Detailed kinematic studies of tearing, as well as improved kinematic models of the requisite internal radula/odontophore components, could attempt to test the anatomical basis for such a functional distinction.

As we stated in the Introduction, kinematic modeling is an iterative activity in which failure is as informative as success. Models are constructed to fit available kinematic data, but also make predictions that can be tested experimentally or validated with higher-resolution data. Our previous kinematic models have led us to the concept of a dynamic radula/odontophore of context-dependent function within a buccal mass whose components are highly interdependent. To explore this concept, we have begun to characterize the detailed three-dimensional kinematics of the radula/odontophore using both in vitro preparations and interleaved mid-sagittal, coronal and axial sections of MR-imaged swallowing adults in vivo(Neustadter et al., 2002b). Such data form the basis for a new generation of kinematic models. Ultimately,we believe that improved kinematic models will allow us to create realistic,force-generating kinetic models that can be used to investigate the neural control of feeding in A. californica. Our goal is a complete description of the interplay between neural control and peripheral biomechanics in this important adaptive behavior.

Movies available on-line.

We thank the Whitehall Foundation (grant M97-12 to H.J.C.) and the NSF(grant IBN-9974394 to H.J.C.; IGERT NSF 9972747) for supporting this research and two anonymous reviewers for helpful comments.

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